Estimating event probability from historical time series with clear seasonality I would like to predict the average number of days in a year for which two conditions are true:


*

*daily average temperature is below zero celsius

*the day was preceded by at least four days with daily average temperature below zero celsius


I've historical daily average temperature data for the location available for about 10 years. My initial approach was to use the one sided Chebyshev inequality
 which can be used to approximate a probability if the distribution is not known. However in this application I am interested in the probability of a special condition, can I use the Chebyshev inequality for a dummy time series as well? I.e.: 1 if condition is fullfilled, otherwise 0, --> the dataset would therefore look something like 0,0,0,0,0,1,1,0,0,0,0,0,1,1,1,1,1,1,etc.
How would you approach a problem like that from a different angle, the data clearly has seasonality, is there any distribution which I could use to have a better estimate than Chebyshev?
 A: I think the joint distribution of temperature data on successive days could be reasonably modelled using a multi-variate Gaussian (Gaussian distributions are often used in statistical downscaling of temperature).  What I would try would be to regress the mean and covariance matrix of the temperature time series on sine and cosine components of the day of year (to deal with the seasonality).  The details on how to do that are given in a paper by Peter Williams, Williams uses a neural network, but I would start off with just a linear model.  This will give you what climatologists would call a "weather generator" (of sorts).  Using this you could generate as many synthetic time series as you want with the appropriate statistical properties, from which you could estimate the probabilities you require directly.  You would need to estimate the window over which temperatures were usefully correllated - which may be quite high in winter due to blocking patterns (for the U.K. anyway).  A bit baroque I suppose, but it would be the thing I would try!
A: I know little about meteorology, so my following assumptions may be wrong: today's temperature is similar to yesterday's and the day before yesterday's (maybe more days going back), and also similar to temperature a year age, two years ago, three years ago, etc.
If these assumptions got reinforcement I would use an ARMA model using days -1, -2, … and -365, -365*2, -365*3, … as predictors of today's temperature, and maybe a few days looking back in the moving average terms. (You can imagine many variants of this model.)
After fitting the model I would make a large number of model based simulations predicting the temperatures for each of the following 365 days, and count the cases satisfying the two conditions.
